 Solution

Part (a) makes ${\lambda }_k<0$ for all sufficiently large $k$, while part (b) would make ${\lambda }_k\ge 0$ whenever $P^{(k)}$ were nonzero. Hence some derivative of $P$ vanishes identically, so $P$ is a polynomial.

Let its degree be $n$. Then $P^{(n)}$ is a nonzero constant. Substituting $k=n$ into the differentiated equation annihilates both derivative terms and givesThus ${\lambda }_n=0$, andThe nonzero regular solutions are therefore constant multiples of the Legendre polynomial of degree $n$.